"Everywhere is here,
once we have shattered
The iron-bound laws of contiguity." Robert Graves, from "Everywhere is here"
(Graves,1966 p. 431)

e have seen that Brunelleschi's peepshow,
by placing the viewer's eye at the center of projection, can give rise to a
compelling illusion of depth. Some students of perspective have thought that
it can also protect the viewer from distortions one might expect to experience
while viewing a picture from a point other than the center of projection.
One of these was Leonardo:

If you want to represent an object
near you which is to have the effect of nature, it is impossible that your
perspective should not look wrong, with every false relation and disagreement
of proportion that can be imagined in a wretched work, unless the spectator,
when he looks at it, has his eye at the very distance and height and direction
where the eye or the point of sight [the center of projection] was placed
in doing this perspective... otherwise do not trouble yourself about it, unless
indeed you make your view at least twenty times as far off as the greatest
width or height of the objects represented, and this will satisfy any spectator
placed anywhere opposite to the picture. (Leonardo da Vinci, 1970, 544, pp.
325-6)

Why did Leonardo expect most paintings to "look wrong" when viewed
from somewhere other than the center of projection? Because he was following
Alberti's prescription of thinking about perspective in geometric terms.
As we saw in Chapter 2, if it is known (or assumed) that a picture such as
Masaccio's Trinity (Figure 2.1) was generated according to the laws of
central projection, it is possible, by making some assumptions about the scene,
to reconstruct the scene.

Fig.5.1 Drawings from La Gournerie, Traité
de perspective linéaire. Panels 95, 96, and 97 represent plans
that solve the inverse perspective problem posed in panel 98. In each
these solutions, ab is picture plane and o is center of projection.
.

Before a geometer can solve the inverse problem of perspective,
the location of the center of projection must be determined. If an error is
made in locating this center, the reconstructed scene will be distorted. For
instance, in Figure 5.1, panel 97 if the center of projection is assumed to
be at point 0. An observer standing at point 0 as specified
in panel 97 would see a rectangular nave, as La Gournerie's plan shows. But
if the center of projection is assumed to have moved to the left, as in panel
96, a geometer cannot solve the inverse perspective problem posed in panel
98 and still reconstruct a building whose ground plan is based on right angles.
The ground plan in panel 96 is a shear transformation of the one
in panel 97: Points of the plan in panel 97 are shifted laterally, parallel
to the picture plane; the greater the distance of a point from the picture
plane, the greater the lateral displacement. (To visualize a shear transformation,
imagine yourself holding a pack of cards and tapping the edge of the pack
against the surface of a table, while the cards are at an oblique angle to
the surface of the table: the pack has undergone ashear from its original shape.) If the assumed center of projection
is moved laterally as much as in panel 96, but is also moved further from
the picture plane, the shear transformation is combined with a magnification,
as shown in panel 95. You may notice that the plan in panel 95 looks less
distorted than the one in panel 96. There are two reasons for this: First,
the amount of shear is smaller in panel 95; p¢ is closer to p in panel
95 than in panel 96. Second, the greater the magnification, the smaller the
angle at which the nave intersects the picture plane.

If perception solved the problem of inverse perspective in the same way as
the geometer would, and if perception assumed that the center of projection
always coincides with the perceiver's current point of view,1 then
an observer standing at point O as specified in panels 95 and 96
would see an oblique nave in accord with La Gournerie's plan.2
As the reader can ascertain by moving in front of panel 98, no such striking
distortions are experienced. I call this violation of our geometric expectations
by our perceptual experience the robustness of perspective.

Such claims about the robustness of perspective have been made before, but
not everyone agrees with the way the problem has been formulated and about
the nature of the evidence in favor of robustness. For instance, Rosinski
and Farber write:

Virtually every writer on pictorial distortion (the
present ones included) has appealed to the reader's intuitions. For example,
Haber (1978, p. 41) in discussing expected perceptions of distorted pictorial
space argues that "most picture lookers know that this does not happen."
It is worth pointing out that neither such casual phenomenology nor the more
experimental phenomenology of Pirenne is relevant here. The fact that observers
are not consciously aware of distortions in virtual space [the depicted space]
does not imply that the nature of virtual space is unregistered by the visual
system. Furthermore, one's introspections about the nature of perceptual distortions
are irrelevant. To comment on whether a picture seems distorted is to assess
a correspondence between virtual space and the represented scene. A judgment
of a distortion of space implies that virtual space is registered and somehow
compared to environmental space. But, observers cannot judge that a scene
is distorted unless they know what it is supposed to look like. This information
is not available at the incorrect viewing point. Logically, one's estimate
of the distortion present in virtual space can not be accurate unless an impossible
object results. (1980, p. 150)

1 Assuming for the sake of simplicity that the
perceiver's point of view is at a point.

2 In recent years, several scholars have presented geometric analyses
of the expected effects of viewing a perspective picture from a point other
than the center of projection: Adams (1972), Farber and Rosinski (1978), Lumsden
(1980), and Rosinski and Farber (1980). As far as I can tell, their only advantages
over La Gournerie's analysis are their accessibility and their occasional
pedagogical felicities.